Computational geometry

Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of self-interested agents who want to form a network connecting certain endpoints, the set of stable solutions—the Nash equilibria—may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context—it is the optimal solution that can be proposed from which no user will defect. We consider the price of stability for network design with respect to one of the most widely studied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fair-division scheme can be derived from the Shapley value and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is $O(\log k)$, where $k$ is the number of users, and that a good Nash equilibrium can be achieved via best-response dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form near-optimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of best-response dynamics, and discuss extensions to a weighted game.

/pdf/the-price-of-stability-for-network-design-with-fair-cost-89te3x19ow.pdf

The Price of Stability for Network Design with Fair Cost Allocation

In this chapter, you will see how the simplex method simplifies when it is applied to a class of optimization problems that are known as “network flow models.” You will also see that if a network flow model has “integer-valued data,” the simplex method finds an optimal solution that is integer-valued.

Flows in Networks

Journal of the ACM

Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of self-interested agents who want to form a network connecting certain endpoints, the set of stable solutions - the Nash equilibria - may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context - it is the optimal solution that can be proposed from which no user will "defect". We consider the price of stability for network design with respect to one of the most widely-studied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fair-division scheme can be derived from the Shapley value, and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is O(log k), where k is the number of users, and that a good Nash equilibrium can be achieved via best-response dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form near-optimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of best-response dynamics, and discuss extensions to a weighted game.

/pdf/the-price-of-stability-for-network-design-with-fair-cost-4b3bbv50mq.pdf

The price of stability for network design with fair cost allocation

This paper concerns graph spanners that are resistant to vertex or edge failures. In the failure-free setting, it is known how to efficiently construct a $(2k-1)$-spanner of size $O(n^{1+1/k})$, and this size-stretch trade-off is conjectured to be tight. The notion of fault tolerant spanners was introduced a decade ago in the geometric setting [C. Levcopoulos, G. Narasimhan, and M. Smid, in Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 186-195]. A subgraph $H$ is an $f$-vertex fault tolerant $k$-spanner of the graph $G$ if for any set $F\subseteq V$ of size at most $f$ and any pair of vertices $u,v\in V\setminus F$, the distances in $H$ satisfy $\delta_{H\setminus F}(u,v)\leq k\cdot\delta_{G\setminus F}(u,v)$. A fault tolerant geometric spanner with optimal maximum degree and total weight was presented in [A. Czumaj and H. Zhao, Discrete Comput. Geom., 32 (2004), pp. 207-230]. This paper also raised as an open problem the question of whether it is possible to obtain a fault tolerant spanner for an arbitrary undirected weighted graph. The current paper answers this question in the affirmative, presenting an $f$-vertex fault tolerant $(2k-1)$-spanner of size $O(f^{2}k^{f+1}\cdot n^{1+1/k}\log^{1-1/k}n)$. Interestingly, the stretch of the spanner remains unchanged, while the size of the spanner increases only by a factor that depends on the stretch $k$, on the number of potential faults $f$, and on logarithmic terms in $n$. In addition, we consider the simpler setting of $f$-edge fault tolerant spanners (defined analogously). We present an $f$-edge fault tolerant $(2k-1)$-spanner with edge set of size $O(f\cdot n^{1+1/k})$ (only $f$ times larger than standard spanners). For both edge and vertex faults, our results are shown to hold when the given graph $G$ is weighted.

/pdf/fault-tolerant-spanners-for-general-graphs-400vxf0dlo.pdf

Fault Tolerant Spanners for General Graphs

The rules governing the availability and quality of connections in a wireless network are described by physical models such as the signal-to-interference & noise ratio (SINR) model. For a collection of simultaneously transmitting stations in the plane, it is possible to identify a reception zone for each station, consisting of the points where its transmission is received correctly. The resulting SINR diagram partitions the plane into a reception zone per station and the remaining plane where no station can be heard.SINR diagrams appear to be fundamental to understanding the behavior of wireless networks, and may play a key role in the development of suitable algorithms for such networks, analogous perhaps to the role played by Voronoi diagrams in the study of proximity queries and related issues in computational geometry. So far, however, the properties of SINR diagrams have not been studied systematically, and most algorithmic studies in wireless networking rely on simplified graph-based models such as the unit disk graph (UDG) model, which conveniently abstract away interference-related complications, and make it easier to handle algorithmic issues, but consequently fail to capture accurately some important aspects of wireless networks.The current paper focuses on obtaining some basic understanding of SINR diagrams, their properties and their usability in algorithmic applications. Specifically, based on some algebraic properties of the polynomials defining the reception zones we show that assuming uniform power transmissions, the reception zones are convex and relatively well-rounded. These results are then used to develop an efficient approximation algorithm for a fundamental point location problem in wireless networks.

SINR diagrams: towards algorithmically usable SINR models of wireless networks

The rules governing the availability and quality of connections in a wireless network are described by physical models such as the signal-to-interference & noise ratio (SINR) model. For a collection of simultaneously transmitting stations in the plane, it is possible to identify a reception zone for each station, consisting of the points where its transmission is received correctly. The resulting SINR diagram partitions the plane into a reception zone per station and the remaining plane where no station can be heard. 
SINR diagrams appear to be fundamental to understanding the behavior of wireless networks, and may play a key role in the development of suitable algorithms for such networks, analogous perhaps to the role played by Voronoi diagrams in the study of proximity queries and related issues in computational geometry. So far, however, the properties of SINR diagrams have not been studied systematically, and most algorithmic studies in wireless networking rely on simplified graph-based models such as the unit disk graph (UDG) model, which conveniently abstract away interference-related complications, and make it easier to handle algorithmic issues, but consequently fail to capture accurately some important aspects of wireless networks. 
The current paper focuses on obtaining some basic understanding of SINR diagrams, their properties and their usability in algorithmic applications. Specifically, based on some algebraic properties of the polynomials defining the reception zones we show that assuming uniform power transmissions, the reception zones are convex and relatively well-rounded. These results are then used to develop an efficient approximation algorithm for a fundamental point location problem in wireless networks.

SINR Diagrams: Towards Algorithmically Usable SINR Models of Wireless Networks

For a set S of points in ℝd, a t-spanner is a sparse graph on the points of S such that between any pair of points there is a path in the spanner whose total length is at most t times the Euclidean distance between the points. In this paper, we show how to construct a (1 + e)-spanner with O(n/ed) edges and maximum degree O(1/ed) in time O(n log n). A spanner with similar properties was previously presented in [6, 8]. However, using our new construction (coupled with several other innovations) we obtain new results for two fundamental problems for constant doubling dimension metrics: The first result is an essentially optimal compact routing scheme. In particular, we show how to perform routing with a stretch of 1 + ∈, where the label size is [log n] and the size of the table stored at each point is only O(log n/ed). This routing problem was first considered by Peleg and Hassin [11], who presented a routing scheme in the plane. Later, Chan et al. [6] and Abraham et al. [1] considered this problem for doubling dimension metric spaces. Abraham et al. [1] were the first to present a (1 + ∈) routing scheme where the label size depends solely on the number of points. In their scheme labels are of size of [log n], and each point stores a table of size O(log2 n/ed). In our routing scheme, we achieve routing tables of size O(log n/ed), which is essentially the same size as a label (up to the factor of 1/ed). The second and main result of this paper is the first fully dynamic geometric spanner with poly-logarithmic update time for both insertions and deletions. We present an algorithm that allows points to be inserted into and deleted from S with an amortized update time of O(log3 n).

/pdf/improved-algorithms-for-fully-dynamic-geometric-spanners-and-17aaumc1se.pdf

Improved algorithms for fully dynamic geometric spanners and geometric routing

An f-sensitivity distance oracle for a weighted undirected graph G(V, E) is a data structure capable of answering restricted distance queries between vertex pairs, ie, calculating distances on a subgraph avoiding some forbidden edges This paper presents an efficiently constructible f-sensitivity distance oracle that given a triplet (s, t, F),where s and t are vertices and F is a set of forbidden edges such that |F| ≤ f, returns an estimate of the distance between s and t in G(V, E\ F) For an integer parameter k ≥ 1, the size of the data structure is O(fkn1+1/k log (nW)), where W is the heaviest edge in G, the stretch (approximation ratio) of the returned distance is (8k-2)(f +1), and the query time is O(|F|ċlog2nċlog log nċlog log d), where d is the distance between s and t in G(V, E\F)

The paper also considers f-sensitive compact routing schemes, namely,routing schemes that avoid a given set of forbidden (or failed) edges It presents a scheme capable of withstanding up to two edge failures Given a message M destined to t at a source vertex s, in the presence of a forbidden edge set F of size |F| ≤ 2 (unknown to s), our scheme routes M from s to t in a distributed manner, over a path of length at most O(k) times the length of the optimal path (avoiding F) The total amount of information stored in vertices of G is O(kn1+1/k log (nW) log n)

https://www.openu.ac.il/home/mikel/papers/FT_DO.pdf

Liam Roditty

Papers

Fault Tolerant Spanners for General Graphs

SINR diagrams: towards algorithmically usable SINR models of wireless networks

SINR Diagrams: Towards Algorithmically Usable SINR Models of Wireless Networks

Improved algorithms for fully dynamic geometric spanners and geometric routing

f-sensitivity distance Oracles and routing schemes